Cascadic Multigrid Method for The Elliptic Monge-Ampere Equation

The elliptic Monge-Ampere (M-A) equation is a fully nonlinear partial differential equation, which originated in geometric surface theory and has been widely applied in dynamic meteorology, elasticity, geometric optics, image processing and others. The numerical solution of the elliptic Monge-Ampere equation has been a subject of increasing interest recently. In this paper, the cascadic multi-grid method (CMG) is used to solve numerically the M-A equation. Before the application of CMG method, an equivalent form of M-A equation is given. On each successive refinement level, weak formulation of this equivalent form can be written and finite element methods can be used successfully. We analyze the convergence and computational complexity for the cascadic multigrid method. And we find that the CMG method is optimal with respect to the energy norm. Finally, numerical experiments confirm the efficiency and robustness of CMG method.